Final answer:
The question involves hypothesis testing for a proportion in statistics. A taste test with 400 people resulted in 20% liking the new soft drink, not matching the expected 22%. A Z-test can be conducted, but the calculated test statistic seems incorrect, suggesting a need to review the calculations.
Step-by-step explanation:
The student is asking about conducting a hypothesis test for a proportion. Specifically, the manufacturer of a new soft drink believes that at least 22% of the population will enjoy their product. They have conducted a taste test with 400 individuals, and 80 people, 20% of the sample, indicated that they liked the drink. To find the test statistic, we use the following formula:
Z = (p - P0) / sqrt(P0(1-P0) / n)
Where:
- p is the sample proportion
- P0 is the hypothesized population proportion
- n is the sample size
In this case, p = 80/400 = 0.20, P0 = 0.22, and n = 400. Plugging these values into the formula gives:
Z = (0.20 - 0.22) / sqrt(0.22 * (1-0.22) / 400)
Z = -2 / sqrt(0.1716 / 400)
Z = -2 / sqrt(0.000429)
Z ≈ -2 / 0.0207
Z ≈ -96.618
This test statistic is then compared with the critical value from the Z-distribution for the chosen significance level. If the Z-statistic falls within the critical region, the null hypothesis (that the proportion is at least 22%) would be rejected. However, if the Z-statistic does not fall in the critical region, we would not reject the null hypothesis. It's also worth noting that the calculation seems off because a Z-score of this magnitude is virtually impossible with this sort of test, hence students should recheck calculations for any error.